reservoir simulation, 5775_chap01

LECTURE NOTES ON APPLIED RESERVOIR SIMULATION © World Scientific Publishing Co. Pte. Ltd.http://www.worldscibooks.com/environsci/5775.html

July 5, 2005 14:55 WSPC/Book-SPI-B283 (9in x 6in) Lecture Notes in Applied Reservoir ch01

Chapter 1

INTRODUCTION

Reservoir simulation, or modeling, is one of the most powerful

techniques currently available to the reservoir engineer. Mod-eling requires a computer, and compared to most other reser-

voir calculations, large amounts of data. Basically, the modelrequires that the field under study be described by a grid sys-

tem, usually referred to as cells or gridblocks. Each cell must

be assigned reservoir properties to describe the reservoir. Thesimulator will allow us to describe a fully heterogeneous reser-

voir, to include varied well performance, and to study differentrecovery mechanisms. Additionally, due to the amount of data

1



2 Lecture Notes in Applied Reservoir Simulation

required, we often will reconsider data which had previouslybeen accepted. To make the model run, we perturb the system

(usually by producing a well) and move forward in time usingselected time intervals (timesteps) as indicated in Fig. 2.5, p.

9, M-13. The main type of results that we gain from a model

study are saturation and pressure distributions at various timesas shown on p. 8, M-13; quite frequently, these variations will

indicate what the primary drive mechanism is at any given pointin time.

On the other hand, modeling requires a computer with afair amount of memory and a great deal of engineering time;

you cannot do a model study in an afternoon! It takes time tolocate the data, modify it to fit your grid system, enter it and

then to actually run the model. Minimum time for a very simplestudy is a week; average time is probably from 3 to 6 months;

large and/or complex studies may encompass years. In short, ittakes much more effort on your part to interpret the results of a

simulator and as a result, small screening models may be usedto evaluate key parameters while larger models would simulate

an entire field in detail. As the field is developed and more data

becomes available, intermediate models are often developed forspecific regions or recovery processes in the field; these models

may be called scalable models, but changing the grid presentsadditional problems.

To be able to decipher what the model is telling you, youmust first define the problem. Simply running a study to model

a field is not good enough; you must decide ahead of timewhat questions you are trying to answer. Some typical questions

might be:

• What type of pattern should be used for water injection?• Should a well be drilled in a certain location?• How would rate acceleration affect the ultimate recovery?• What is the effect of well spacing?



Introduction 3

• Is there flow across lease lines?• Will the oil rim rise to saturate the gas cap?• Should gas injection be considered? If so, for how long?• Should water injection be considered? If so, at what rates?

Once we have decided what questions need to be answered,

we can construct the model grid.

1.1. Types of Models

There are five types of models, depending on the grid selected,that may be used (although the first two types are used mini-

mally today):

• One-dimensional horizontal• One-dimensional vertical• Areal (two-dimensional)• Cross-sectional (two-dimensional)• Three-dimensional

Model grid — Cartesian coordinate system




Various grid systems are illustrated on p. 7 in M-13. Addition-ally, the coordinate system, number of components (or phases)

and treatment of the flow equations yield a large number of sim-ulation possibilities. The most common coordinate system in use

is that of cartesian (rectangular) coordinates.

A one-dimensional (1-D) model may be used to definea bottom water drive, determine aquifer activity, yield an accu-

rate material balance or as a screening tool prior to a largecomplex study. Gravity drainage may be simulated using a 1-D

vertical model. Sensitivity studies may be conducted and inter-preted rapidly using 1-D models; these studies might include

the effects of vertical permeability, injection rate, relative per-meability, residual oil saturation, reservoir size, etc. This infor-

mation would be extremely useful in more complex studies.Individual well behavior cannot be modeled using a 1-D model;

however, field behavior may be approximated. Trying to matchproduction history of individual wells using a 1-D model is both

fruitless and time consuming. 1-D models are seldom used exten-sively today.

There are two types of two-dimensional (2-D) cartesian

models; the most common is the areal model. Strictly speaking,

One-dimensional models



Introduction 5

an areal model should be used only if there will be very littlevertical movement of fluids as in a thin sand; however, the areal

model is also employed for thick sands when no great differ-ences in permeability exist (i.e., permeability layering). Dip can

be incorporated in an areal model, although water underrunning

or gas overriding may not be in its proper perspective if perme-ability layering exists. The effects of varying well patterns, both

in type and spacing may be studied with an areal model.

Areal model

The other type of 2-D cartesian model, the cross-sectionalmodel, is often used to simulate a slice of a field. It will show

vertical and horizontal movement, but is not useful for determin-ing well patterns. Its greatest usage is in determining comple-

tion intervals and stratification effects. Usually, when orientinga cross-sectional model (commonly called an X-Z model), the

cross-section is taken parallel to the fluid movement (up or downdip). This type of model is used for thick, layered reservoirs,

water underrunning, gas segregation, or a series of reservoirs

co-mingled in the wellbore.




Cross-sectional model

The three-dimensional (3-D) model can handle any and

all of the previous types of studies; however, the computer time

and interpretive engineering time are greatly increased over thatrequired for 2-D models. A 3-D model must be used when fluid

migration is expected parallel to the strike of a thick steeplydipping bed (i.e., fluids will flow up dip and across dip). If a

typical section of a field cannot be determined for use in a 2-Dmodel, then a 3-D model is required; however, finely modeling

Three-dimensional model



Introduction 7

the area of concern and “lumping” the remainder of the field intoa few large cells may save considerable time and money as shown

in the windowed model (Fig. 3.29, p. 24, M-13). Once again, youmust define your problem before you start to model it.

The second type of coordinates employed in simulation is

the radial (R-Z-Θ) or cylindrical system and may exist in oneto three dimensions. Radial systems in two dimensions (R-Z)

are sometimes referred to as coning models based on theirearly applications for studying the effects of coning phenom-

ena. They are single well models designed to study individualwell effects; additional wells may be included, but they will not

exhibit the performance shown in actual production. Coningmodels are fully implicit in order to handle the rapid satura-

tion changes that occur near the wellbore. Field studies (wholeor partial) may also be performed using a cylindrical system, but

this application has found limited use. Aquifers may be simu-lated in radial models by use of a water injection well in the

outer block; this technique works well for strong aquifers butmay present problems with weaker water drives. Radial models

may be used to study coning, shale breaks, well tests, vertical

R

θZ

Radial coordinate system




permeability effects, heterogeneity, and to determine maximumproducing rates; however, when studying coning, after shut-in,

the cone will fall in a simulator without hysteresis; whereas inreality, the cone will not completely drop and imbibition effects

will greatly inhibit future production; this concept is discussed

in Chaps. 4 and 9 of this book.

Two-dimensional radial model

Black oil (or Beta) models consist of three phase flows:

oil, gas, and water, although additional gas or aqueous phasesmay be included to allow differing properties. These models

employ standard PVT properties of formation volume factorsand solution gas and are the most common type of simulator.

PVT properties are covered later in this chapter and in Chap. 3.Compositional simulators are similar to black oil models

as far as dimensions and solution techniques are concerned; here,the similarity ceases, for while volume factors and solution gas

effects are employed in a black oil model, a compositional modelemploys Equations of State (EOS) with fugacity constraints,

and uses equilibrium values, densities and several varying com-

ponents (including non-hydrocarbons). Considerable time is



Introduction 9

required in the phase package (i.e., matching lab data with sim-

ulator requirements) before the actual model can be run. It isreasonable to state that this type of model requires additional

expertise to be useful.

Finally, treatment of the model equations yields either anIMPES (implicit pressure, explicit saturation) formula-

tion, a fully implicit formulation, or some combination thereof.Very simply, an IMPES model is current in pressure and solves

for saturations after pressures are known while a fully implicitmodel solves for both pressures and saturations simultaneously.

Rapid saturation changes require fully implicit models. Thesemi-implicit treatment is a combination which attempts to esti-

mate what saturations will exist at the end of the timestep.

1.2. Data Requirements

Variables required for assignment to each cell (locationdependent):

• Length• Width• Thickness• Porosity• Absolute permeabilities (directional)• Elevation• Pressure(s)• Saturations

Variables required as a function of pressure:

• Solution gas–oil ratio• Formation volume factors• Viscosities• Densities• Compressibilities




Variables required as a function of saturation:

• Relative permeability• Capillary pressure

Well data:

• Production (or injection) rate• Location in grid system• Production limitations

A similar but more confusing outline of the data required formodeling is on p. 30, M-13.

∆x1

i-1,j,k

x

yz

(1,1,1)

i,j,k

(2,1,1)

i+1,j,k

(3,1,1)

∆x2 ∆x3

∆z

∆y

Directional notation

Lengths are normally obtained by superimposing a grid sys-tem on a field map and measuring the appropriate distances.

These increments are usually denoted using the variable ∆xwith the subscript “i” referring to the cell location by column

(running from left to right). The standard practice of overlaying

a grid on a map is used for one-dimensional (both horizontal



Introduction 11

and vertical), areal and three-dimensional models. For dipping

reservoirs, the aerial distances will be shorter than the actual

distances between the wells. Usually, this discrepancy is notapparent due to the available accuracy of several of the reser-

voir descriptive parameters, particularly for dip angles of lessthan 10◦; however, the variation may be corrected using pore

volume and transmissibility modifiers or as an input option insome simulators. The actual length is r = x/cos Θ.

Widths are measured in the same manner as lengths andthe same discussion applies. Note that the widths in a cross-

sectional model need not be constant. Widths are denoted as∆y with a subscript “j” and are sequenced by rows from rear

to front (top to bottom in an areal model).Thickness values are obtained from seismic data, net

isopach maps (for areal and 3-D simulations), well records, coreanalysis and logs (for cross-sectional models). Thicknesses in an

areal model may vary with each cell and are denoted as ∆z.

For layered models the subscript “k” is employed to denote the




Variable widths in a cross-sectional model

Variable thicknesses in an areal model

layers; they are sequenced from top to bottom. For areal con-siderations (including 3-D), thickness values may be obtained

by superimposing a grid on a net pay isopach. Obviously, thick-ness values may also be obtained by subtracting the bottom of

the formation from the top of formation when these maps are

available; at this point, gross pay is known and must then be



Introduction 13

reduced to net pay. Note that unless a net-to-gross input optionis employed, thickness must be a net pay.

When constructing a cross-sectional model using well recordsand logs, the actual distance between cell centers (centroids) is

employed; however, the pore volumes calculated in this instance

are in error when (vertical) net pay is used since they are calcu-lated based on (length ∗ width ∗ net pay ∗ porosity). Note that

the error introduced tends to compensate for the length errorpreviously discussed.

Dip angle effect on thickness

Porosity (φ) is a ratio of void space per bulk volumeand may be found using logs, laboratory analysis, correlations,

and/or isoporosity contour maps. If thicknesses have alreadybeen determined, porosity values may be calculated from isovol

(φh) maps when available.Total porosity is a measure of total void space to bulk vol-

ume whereas effective porosity is the ratio of interconnectedpore space to bulk volume. For intergranular materials, such as

sandstone, the effective porosity may approach the total poros-

ity; however, for highly cemented or vugular materials, such as




limestones, large variances may occur between effective and totalporosity. In shales, total porosity may approach 40% whereas the

effective porosity is usually less than 2%.Since effective porosity is concerned with the interconnected

void spaces, it should be input to simulators. Note that poros-

ity values obtained from logs (Sonic, Density, or Neutron) willapproach a total porosity value.

Hydrocarbon porosity is a measure of the pore space occu-pied by oil and gas to bulk volume and may be defined as

φh = φ(1 − Sw).

Porosity is independent of rock grain size but is dependent on

the type of packing. A maximum porosity of 47.8% is obtainedfrom cubic packing and a porosity value of 26.0% exists for

rhombohedral packing. In general, porosity values for unfrac-tured systems will range from 0 to 30% with the majority of

values occurring from some minimum value to 20%. Porositiesmay be obtained at either reservoir or a fairly low (∼100 psi)

pressure in the laboratory, although low pressure values are morecommonly reported; log-determined values will be at reservoir

Cubic packing: 47.8%



Introduction 15

Rhombohedral packing: 26.0%

pressure. The effect of pressure on porosity is

φ2 = φ1ecf (p2−p1)

which is sometimes written (using a series expansion) as

φ2 = φ1[1 + cf(p2 − p1)].

Cubic packing — Two grain sizes: 14%




Typical sand

This equation should not be used for extremely soft formations

(∼100 microsips); always use the exponential form of the equa-tion. Note that as pressure decreases, porosity decreases due

to the overburden effect; however, to convert low pressure lab-measured values to reservoir conditions, the pressure change

(p2 − p1) must be reversed to (p1 − p2). Changes in porosity canaccount for compaction in highly compressible formations; com-

paction may or may not be reversible. When averaging porosity

values, use a net pay weighted average:

φavg =n∑

i=1

(φihi)

/n∑

i=1

hi.

Additional information concerning porosity may be found onpp. 29–31, M-13.

Absolute permeability (k or ka) is a measure of the rock

capability to transmit fluids. Absolute permeability has units of



Introduction 17

millidarcies (md) and may be obtained from well tests, labora-tory analysis, correlations or in rare instances, isoperm maps.

Several different techniques are available for analyzing a varietyof well tests. Remember that laboratory results apply only to

the section of core being analyzed while a well test indicates anaverage permeability in a region (usually large) surrounding the

wellbore.Also, well test analyses yield effective permeability val-

ues and the relationship between effective and absolute

permeability is

ke = kakr,

where the relative permeability (kr) is a reduction due to the

presence of other fluids, and will be discussed later in thischapter. Comparisons of core data and well test data are shown

on p. 35, M-13. Often, permeability will correlate with porosity;some sample correlations of permeability as a function of poros-

ity for core data are shown in Figs. 4.10–4.12, pp. 35–36, M-13.Three techniques may be used to calculate average perme-

ability values: arithmetic (or parallel), reciprocal (or series or

harmonic) or geometric averaging.




For cartesian systems having “nz” layers, the arithmeticaverage is

karith =nz∑i=1

(kihi)

/nz∑i=1

hi

which may be used to calculate the horizontal permeability instratified systems.

Parallel averaging

The reciprocal average for cartesian systems with “nx”

columns in series is

krecip =nx∑i=1

Li

/nx∑i=1

Li

ki

which is represented as shown.

A third technique sometimes employed in averaging per-meabilities for randomly distributed data is the geometric

average

kgeo = exp

[(n∑

i=1

hi ln ki

)/n∑

i=1

hi

]



Introduction 19

k1 k2 k3 krecip

LL1 L2 L3

Series averaging

or for “n” evenly spaced intervals,

kgeo = n

√√√√ n∏i=1

ki.

Note that the reciprocal average favors smaller values and that

the geometric average falls somewhere between reciprocal andarithmetic averaging results.

Additionally, permeabilities may have directional trends(anisotropy); for example, in an areal model, the North–South

permeability may be greater than the East–West permeability.In standard cartesian gridding, there may only be two areal per-

meabilities which must be orthogonal and as such, the grid mustbe aligned with any directional trends. In cross-sectional and

3-D models, vertical permeabilities are required; for exam-ple, a sealing shale in a cross-sectional model would have a ver-

tical permeability of zero. Quite frequently, a value of one-tenthof horizontal permeability is used for vertical permeability (note

that this method is not necessarily recommended, only men-

tioned). Both vertical and areal permeability variations may be




determined by well tests. M-13 discusses absolute permeabilityon pp. 31–38.

Elevations (or depths) for areal and 3-D models are usu-ally obtained from structure maps which have been constructed

based on data obtained during drilling and logging as well as

other geological information as shown in Fig. 8.9, p. 96, M-13.The variable used to denote elevations is usually D or h; this

may prove confusing, since h is usually used for net pay (whichis ∆z in most simulators). The simulator requires the elevation

at the centroid of each cell so that top of formation or bottomof formation maps should be adjusted to the center of the cells.

Many simulators will accept top of sand data and adjust it byone-half of the net pay. Elevations may be referenced from any

convenient (and consistent) location: subsea, subsurface (whenhorizontal), kelly bushing, marker sand, or even top or center of

formation. In most models, the directional notation is that down(from the reference elevation) is positive and up is negative. For

smoothly dipping reservoirs, the rate of dip (ft/mile) may becalculated as

5280 tan Θ,

where Θ is the dip angle; often, this calculation is shown as

5280 sinΘ and for dip angles of 10◦ or less, the sine and tangentare numerically similar.

In constructing a cross-sectional model from well records andlogs, the procedure is similar to that described using structure

maps. For layered models (cross-sectional and 3-D), elevationsmay be required for every cell in every layer; when no gross

discontinuities exist, the top layer elevations may be adjustedby averaging the pay zones; however, when the actual reservoir

zones are separated by non-productive rock, elevations must bedetermined for each cell.

Pressures are required for each cell in a simulator and may

be input on a per cell basis; however, if the simulation begins at



Introduction 21

Z1

Z2

D1D1

D2

D2*

D2*=D1 + (∆z1+∆z2)/2

Layer elevation calculations

equilibrium conditions, it is much easier to use a pressure at a

known datum and calculate pressures for all cells using a densitygradient adjustment

P = Pdatum +ρ ∆D

144,

where

P = pressure in cell, psia

Pdatum = datum pressure, psia∆D = change in elevation, ft (+ is down)

ρ = fluid density, lb/ft3.

Additionally, in multiphase flow, a pressure for each phase

(oil, gas and water) must be calculated. The pressure in thewater phase is related to the oil pressure by the capillary pressure

Pw = Po − Pcwo

and the pressure in the gas phase is related to the oil pressure by

Pg = Po + Pcgo.




Saturations (So, Sw, Sg) are also required for each cell; as

with pressures, they may be directly assigned to cells; however, ifthe saturations are known at any given datum (usually the gas–

oil contact and water–oil contact), they may be determined at

equilibrium based on capillary pressures for each cell. For exam-ple, to determine the oil and water saturations 10 feet above

the water–oil contact (defined in this example as 100% water)for a 50 lb/ft3 oil and a 65 lb/ft3 water, the water–oil capillary

pressure, at the contact, is 0 psi (since no oil is present). If thepressure at the WOC is 3000 psi (which is a water pressure),

then

Po = Pw + Pcwo

= 3000 + 0

= 3000 psi.

At a point 10 feet above the WOC, the oil pressure is

Po = Po datum + ρo∆D/144

= 3000 + (50)(−10/144)

= 3000 − 3.5

= 2996.5 psi

and the water pressure is

Pw = Pw datum + ρw∆D/144

= 3000 + (65)(−10/144)

= 3000 − 4.5

= 2995.5 psi

and the capillary pressure 10 feet above the WOC is

Pcwo = Po − Pw

= 2996.5 − 2995.5

= 1.0 psi



Introduction 23

so the water saturation at this point corresponds to the valuewhich exists at a capillary pressure of 1 psi. This same technique

is explained in Sec. 4.7.1 on p. 41, M-13 and will make a lot moresense after the discussion on capillary pressure at the end of this

chapter.

Solution gas–oil ratio (Rs) or dissolved gas is required asa function of pressure and based on the pressure in each cell, the

amount of solution gas will be calculated for each cell. It mayhave units of either SCF of solution gas per STB oil, or MCF

solution gas per STB oil; in the former case, the values shouldbe between 50 and 1400 SCF/STB with the majority of fields

falling between 200 and 1000 over reasonable pressure ranges.Obviously, for units of MCF/STB, the variations are 0.05 to

1.4, etc. Quite frequently, dissolved gas values are given with-out units and it is necessary to determine the appropriate units.

When plotted as a function of pressure, solution gas remainsconstant above the bubble point and decreases with decreasing

pressure below the bubble point as gas is released from solu-tion to become free gas. Although curvature exists below the

bubble point, a large number of solution gas samples exhibit

a markedly linear relationship, and a reasonable first-guess can

Solution gas plot




often be obtained by using the bubble point value and a dead-oilvalue of zero at atmospheric pressure.

Two types of liberation processes may be used to measuresolution gas: flash and differential. In a flash liberation pro-

cess, gas which is released from solution remains in contact with

the oil (a constant composition process) whereas in differentialliberation, the free gas is removed while maintaining pressure.

Flow in reservoirs with any appreciable vertical permeability willapproximate a differential process while tubing, surface equip-

ment and reservoirs having continuous shales approach a flashprocess. Laboratory analyses usually give pressure-dependent

differential values of solution gas and a bubble point flash value;pressure-dependent flash values may be calculated using

Rsflash= Rsdifferential

Rsbpflash

Rsbpdifferential

.

Flash and differential liberation

Most simulation studies will have solution gas values avail-able from a laboratory analysis; however, for some preliminary

studies, it may be necessary to estimate dissolved gas using cor-relations.



Introduction 25

Solution gas–water ratio or the dissolved gas in water isrequired in some models. While the same concept as for dissolved

gas in oil applies, the amount of gas soluble in most aquifers issignificantly less, ranging from 4 to 20 SCF/STB; Rsw is the

variable used to denote dissolved gas in water. In general, for oil

and gas simulations, omitting the effects of Rsw causes no visiblechange in the results.

Oil formation volume factors (Bo) relate a reservoir vol-ume of oil to a surface volume. The reservoir volume includes

dissolved gas whereas the surface volume does not. The oil for-mation volume factor has units of RVB/STB. A reasonable

range for the oil formation volume factor is from 1.05 to 1.40RVB/STB. Note that the oil formation volume factor includes

any dissolved gas; very simply, dissolved gas is considered aspart of the oil. Below bubble point pressure, a decrease in pres-

sure results in a decrease in Bo due to the fact that dissolvedgas is released from the oil yielding a lesser volume at the lower

pressure.

Oil formation volume factor plot

Above the bubble point (in an undersaturated condition), a

decrease in pressure releases no solution gas and when relieving

the pressure on a fixed volume, expansion occurs, and the oil




Flash and differential plot

formation volume factor increases (slightly) with a decrease in

pressure until the bubble point is reached.Both flash and differential liberation techniques are used in

the laboratory for determining oil formation volume factors andthe discussion given for solution gas also applies to the oil for-

mation volume factor. Flash values of the oil formation volumefactor may be determined from

Boflash= Bodifferential

Bobpflash

Bobpdifferential

.

Use of flash data may cause severe timestep limitations when

going through the bubble point.The oil formation volume factor is usually a gentle curve up

to the bubble point and over a limited pressure range is a fairlystraight line above the bubble point. Above the bubble point,

Bo = Bobp e−co(P−Pbp)

which is often shown using a power series expansion as

Bo = Bobp[1 − co(P − Pbp)],



Introduction 27

where

Bo = oil formation volume factor, RVB/STB (above bubblepoint)

Bobp = oil formation volume factor, RVB/STB (at bubblepoint)

co = undersaturated oil compressibility, psiP = reservoir pressure, psia

Pbp = bubble point pressure, psia.

For highly undersaturated reservoirs, use the exponential form

of the oil formation volume factor equation. Note that the oilformation volume factor above the bubble point must always be

less than the bubble point value.Gas formation volume factor (Bg) is a function of pres-

sure; unfortunately, several different units may be applied to

the gas formation volume factor: RCF/SCF, RVB/SCF, orRVB/MCF. Since many flow rates are measured in MCF/day

and the combination of rate times volume factor is desired, anddue to the fact that the values are in the range of 0.1 (at high

pressures) to 35 (at low pressures), RVB/MCF is a preferredset of units. For most reservoir pressures encountered, Bg will

Gas formation volume factor plot




be between 0.2 and 1.5 RVB/MCF. The gas formation volume

factor is readily calculated from

Bg =5.035z(T + 460)

P,

where

Bg = gas formation volume factor, RVB/MCF

z = gas deviation factorT = reservoir temperature, F

P = reservoir pressure, psia.

The gas formation volume factor increases with decreasing pres-sure due to expansion. Values of the gas deviation factor (z-

factor) may be obtained from laboratory analysis of gas samplesor correlations such as the z-factor chart by Standing and Katz

or the resultant equations of Yarborough and Hall, or others.

Water formation volume factors (Bw) are required as afunction of pressure although many simulators employ a value

at a base pressure and correct it using

Bw = Bwbe−cw(P−Pb) ≈ Bwb[1 − cw(P − Pb)],

where

Bw = water formation volume factor, RVB/STB

Bwb = water formation volume factor at Pb, RVB/STBcw = water compressibility, /psi

P = reservoir pressure, psiPb = base pressure, psi.

Water formation volume factors are usually very close to 1.0,

ranging from 1.0 to 1.05 RVB/STB. Due to the small amount ofgas dissolved in water, the formation volume factor will increase

slightly with decreasing pressure. Water formation volume factordata is seldom available from the lab and correlations are usually

employed; this is due to the fact that the slight deviation from

1.0 usually does not warrant the expenditure for a lab analysis.



Introduction 29

Oil viscosity (µo) is a measure of the molecular interaction

(the intertwining of hydrocarbon chains) and is required as afunction of pressure in simulators; standard units are centipoise

(cp). Frequently, it is available from laboratory analyses, either

at a base pressure or reservoir pressures. If unavailable, it maybe estimated (or corrected from base pressure to reservoir condi-

tions) from correlations. Oil viscosity increases with decreasingpressure at saturated conditions (below the bubble point) due

to the release of solution gas (small molecules compared to theoil). Above the bubble point, a decrease in pressure yields a

decrease in oil viscosity because the molecules are not forced asclose together as at the higher pressure.

Oil viscosity plot

Gas viscosity (µg) is primarily a function of pressure; when

measured in the laboratory, it may be reported at a base pres-sure (usually atmospheric) or at reservoir pressures. As pressure

decreases, gas viscosity decreases. A reasonable range of gas vis-cosity values is from 0.01 to 0.04 cp with higher values at pres-

sures in excess of 10,000 psi. When unavailable as laboratorydata, gas viscosities may be estimated using correlations.

Water viscosity (µw) is seldom input to simulators at vary-

ing pressures due to the fact that it is somewhat independent




Gas viscosity plot

of pressure being primarily a function of temperature and to a

lesser degree, a function of salinity. Sometimes a base pressureand reservoir temperature value is available from lab analysis

when required; if not, a correlation may be employed. A normalrange for water viscosities at reservoir temperatures is from 0.3

to 0.8 cp.Oil density (ρo) is almost always reported in terms of a

stock tank gravity (which is a dead oil); most simulators adjustthis value to reservoir conditions using the following relationship

below the bubble point

ρo =ρoST + 13.56 gg Rs

Bo

,

where

ρo = oil density, lb/ft3

ρoST = stock tank oil density, lb/ft3

gg = gas gravity

Rs = dissolved gas, MCF/STBBo = oil formation volume factor, RVB/STB.

Above the bubble point, RsBP is used in place of Rs.



Introduction 31

Oil densities are normally reported as API gravities and therelationship between API gravity and density in lb/ft3 is

ρoST =(62.4)(141.5)

131.5 + API=

8829.6

131.5 + API.

Note that the oil densities will be used to determine the pres-sure gradients for initialization in the simulator using ρo/144 to

obtain the gradient in psi/ft. A normal range of API gravities

is from 45◦ to 10◦ corresponding to densities of 50.0 and 62.4 inlb/ft3 respectively.

Gas density (ρg) is usually input as a gas gravity (gg orγg) or in units of lb/MCF. The relationship between these two

quantities at standard conditions is

ρgST =(28.9)(14.7)(1000) gg

(10.73)(460 + 60)= 76.14 gg,

where

ρgST = gas density, lb/MCF

gg = gas gravity

and gas densities are generated from

ρg =1000ρgST

5.615Bg

,

where

Bg = gas formation volume factor, RVB/MCF

and density gradients are calculated with ρg/144000 in psi/ft.

A normal range for gas gravities is from 0.6 to 1.2 which corre-

sponds to values of 45.7 to 91.4 in lb/MCF.




Water density (ρw) is required as either a density in lb/ft3

or as a specific gravity (γw). The relationship between the two is

ρw = 62.4 γw

and due to salinity the water density at standard conditions maybe estimated from

ρwST = 62.4 + 0.465S,

where

S = salinity, %.

Finally, the standard density may be corrected to reservoir con-

ditions using

ρw =ρwST

Bw

and the gradient calculated from ρw/144. Most oilfield watershave densities slightly greater than 62.4 lb/ft3.

Oil compressibility (co) may be defined either above orbelow the bubble point; however, the only value(s) required in

simulators are for undersaturated conditions where the com-pressibility is used to adjust the oil formation volume factor

from bubble point conditions, using either

Bo = Bobp e−co(P−Pbp)

or

Bo = Bobp[1 − co(P − Pbp)]

as shown earlier. Oil compressibility may be measured in the

laboratory or obtained from correlations. Standard units for oilcompressibility are /psi which yields oil compressibility values

ranging from 6 × 10−6 to 20 × 10−6; a more recent unit is themicrosip which is 106 times greater.

Water compressibility (cw) is almost always obtainedfrom correlations. For undersaturated conditions, it is usually



Introduction 33

a number close to 3 × 10−6/psi (or 3 microsips) at reservoirconditions.

Formation compressibility (cf), sometimes mistakenlyreferred to as rock compressibility, is primarily a measure of

the pore volume compression of the formation. Data are seldomavailable and correlations are often employed. The usual range

of formation compressibilities (for hard reservoir rocks) is from3 × 10−6 to 8 × 10−6/psi although some limestones may exhibit

higher values at low porosity.Relative permeability (kr) is a reduction in flow capability

due to the presence of another fluid and is based on

• pore geometry• wettability• fluid distribution• saturation history.

Relative permeability effect

Relative permeability is dimensionless and is used to deter-mine the effective permeability for flow as follows:

ke = kakr.




Relative permeability data are entered in models as functions ofsaturation and may be obtained from laboratory measurements,

field data, correlations, or simulator results of a similar forma-tion. Whether appropriate or not, it is usually the first data to

be modified in a model study. The simplest concept in relative

permeability is that of two-phase flow. For oil reservoirs, thecombinations are water–oil and liquid–gas (usually thought of

as oil–gas); for gas reservoirs, gas–water applies; and for con-densate reservoirs, gas–liquid.

Water–oil relative permeability is usually plotted as afunction of water saturation. At the critical (or connate) water

saturation (Swc), the water relative permeability is zero

krw = 0

and the oil relative permeability with respect to water (or, in

Water–oil relative permeability

the presence of water) is some value less than one

krow < 1.0.

At this point, only oil can flow and the capability of the oil toflow is reduced by the presence of critical water. Note that data

to the left of the critical water saturation is useless (unless the

critical water becomes mobile). As water saturation increases,



Introduction 35

the water relative permeability increases and the oil permeabil-ity (with respect to water) decreases. For the oil reservoir proper,

a maximum water saturation is reached at the residual oil sat-uration (Sorw); however, since models use an average saturation

within each cell, oil saturation values of less than residual oil (in

a cell) should be correctly entered. Also, the end point value ofkrw = 1 at Sw = 1 would be required when an aquifer is being

included in the simulation study.Wettability is a measurement of the ability of a fluid to coat

the rock surface. Classical definitions of wettability are based onthe contact angle of water surrounded by oil and are defined as

Θ < 90◦ water–wet

Θ > 90◦ oil–wet

Θ = 90◦ intermediate or mixed wettability.

Contact angles

A variation of up to ±20◦ is usually considered in defining inter-

mediate wettability. Contact angle measurements are difficult to

perform under reservoir conditions.




Unfortunately, there is a second definition of water-oil rel-ative permeability currently in use, known as normalized

relative permeability. This method defines the oil relativepermeability at critical water as having a value of 1 and defines

absolute permeability as the effective permeability with critical

water present. In either case, the effective permeabilities will beidentical. These values of relative permeability may be corrected

to standard values by

krSTD = krNORMkaNORM

kaSTD

where

kaNORM = keo at Swc.

Note that at high water saturations, krw may exceed a value of

one for normalized relative permeability values, particularly foroil-wet systems.

Normalized water–oil relative permeability

Some heuristic rules that may be applied to normalized oil–water relative permeability are shown in the following table;

these rules were originated by Craig and subsequently modifiedby Mohamad Ibrahim and Koederitz (SPE 65631) which would

result in the following types of relative permeability plots based

on wettability.



Introduction 37

Rock Swc Sw at which k∗rw at

Wettability k∗rw and k∗

row Sw = 100 − Sorw

are equal (fraction)

StronglyWater-Wet: ≥15% ≥45% ≤0.07

Water-Wet: ≥10% ≥45% 0.07 < k∗rw ≤ 0.3

Oil-Wet: ≤15% ≤55% ≥0.5

Intermediate: ≥10% 45% ≤ Sw ≤ 55% > 0.3(Mixed-Wet) OR

≤15% 45% ≤Sw ≤ 55% <0.5

Wettability effect on relative permeability




Gas–oil relative permeability, or gas–liquid relative per-meability, is similar in concept to water–oil relative permeability.

The preferred relative permeability values are those taken withcritical water present. As free gas saturation increases, the oil

relative permeability with respect to gas (krog) decreases until

the residual oil saturation with respect to gas (Sorg) is reached;however, until the critical gas saturation (Sgc) is reached, the gas

relative permeability is zero (krg = 0). The critical gas satura-tion is the point at which the gas bubbles become large enough

to break through the oil and away from the rock surface. As gassaturation increases, the gas relative permeability increases and

theoretically reaches a value of unity at 100% gas. In reality, bothin the reservoir and in a simulator, critical water will always be

present, so data to the right of the Swc value are meaningless.Incidentally, usually Sorg < Sorw. Also, note that if krog = 1

at Sg = 0, all values of krog should be multiplied (reduced) bykrow at Swc, but krg values should not be adjusted. A minimal

discussion of relative permeability is on pp. 38–40, M-13.Capillary pressure (Pc) data is required in simulators to

determine the initial fluid distributions and to calculate the pres-

sures of oil, gas and water. It is the difference in pressure betweentwo fluids due to a limited contact environment. This data is

required as a function of saturations and may be obtained from

Gas–oil relative permeability



Introduction 39

laboratory measurements, correlations or estimated to yield thedesired fluid distributions. When laboratory measurements are

used, they must be corrected to reservoir conditions

Pcr = PcLσr

σL,

where

Pcr = capillary pressure at reservoir conditions, psiPcL = capillary pressure at lab conditions, psi

σr = interfacial tension of reservoir fluids, dynes/cmσL = interfacial tension of lab fluids, dynes/cm.

When fluid distributions are known at various depths (from coreanalysis or logging techniques), capillary pressures may be esti-

mated from

Pc =H∆ρ

144,

where

Pc = capillary pressure, psi

H = height of transition zone above denser fluid, ft

∆ρ = difference in density between two fluids, lb/ft3 (a positivenumber).

With rare exceptions (high capillary ranges), capillary pressureshave minimal effects once the reservoir is produced.

Water–oil capillary pressure may be determined fromeither of the two techniques previously examined. It ranges from

0 psi at 100% water to a maximum value of between 5 and 25 psi(usually) at critical water. For extremely homogeneous forma-

tions (and from lab data), the (imbibition) curve is as shown;however, for heterogeneous reservoirs, several different sets of

the previous curve will ultimately tend toward a straight line;this same linearity occurs due to gridblock effects as shown in

Fig. 3.22, p. 21, M-13 and as we will see later, due to fluid

segregation.




Homogeneous formation capillary pressure plot

Heterogeneous formation capillary pressure plot

Another correlating factor for water–oil capillary pressure

is the J-function from which capillary pressures may becalculated using

Pc = 4.619J(Sw)σ

√φ

k,

where

Pc = capillary pressure, psi

J(Sw) = J-function value at Sw

σ = interfacial tension, dynes/cm

φ = porosity, fractionk = permeability, md



Introduction 41

Sample J-function plot

and the J-function values are selected at varying water satura-tions from an appropriate correlation as shown. Allowing input

of J-functions in place of capillary pressure data would result

in capillary pressure tables that vary with the porosity and per-meability values assigned to each cell; additionally, varying the

interfacial tension with pressures in the PVT table would allowa capillary pressure variation with pressure.

Gas–oil capillary pressure is usually determined by lab-oratory air–oil data or by estimating the capillary values based

on the height of the transition zone. When using the transitionzone approach, the gas density may be calculated from

ρg = 13.56gg

Bg,

where

ρg = gas density, lb/ft3

gg = gas gravityBg = gas formation volume factor, RVB/MCF.




Gas–oil capillary pressure plot

Since most gas–oil transition zones are short, a reasonable rangeof gas–oil (or gas–liquid) capillary values is from 0 psi at all

liquid (or no free gas) to a maximum value of between 2 and10 psi (usually) at critical water (maximum gas saturation in a

gas cap). The discussion concerning linearity (under water–oilcapillary pressure) is also applicable to gas–oil data.

Production or injection rates are required for each well tobe modeled. For liquids, the rate is usually in STB/day and for

gas, MCF/day. For producing wells, only one phase production

should be specified and that phase is usually the predominantphase. For example, an oil well would specify oil production,

and the appropriate gas and water producing rates would becalculated by the model. This data is normally obtained from

well files; although the data will vary with time, it is acceptableto use an average rate over a given period of time as long as no

drastic rate fluctuation has occurred (see Fig. 7.2, p. 75, M-13).In many cases, a reasonable assumption is to adjust rates when

a variation by a factor of 2 occurs. Average rates should becalculated based on production during the time period

qo =∆Np

∆t,



Introduction 43

where

qo = oil rate, STB/day

∆Np = oil production during time ∆t, days∆t = production time, days.

Well locations in the grid system are also required as shownin Fig. 5.1(b), p. 45, M-13. Remember the cells are numbered

from left to right in the x-direction (the “i” index location),rear to front in the y-direction (the “j” index location), and

top to bottom in the z-direction (the “k” index location). Ingeneral, for areal and 3-D models, a well should be centered in

a cell whenever possible. Vertical layers should correspond tocompletion intervals in cross-sectional and 3-D models as shown

in Fig. 7.5, p. 80, M-13.Production limitations may be imposed on wells. Some

of these may be bottom-hole pressures, skin factors, maximum

GOR or WOR limits, total field limitations, coning effects andabandonment conditions (rate, GOR, WOR). Although called

production limits, many of these conditions may also be appliedto injectors. The end result of all of these stipulations is to adjust

the rate data somewhat automatically. A brief discussion of wellmanagement may be found in Sec. 2.6 (p. 11), of M-13 and a

more complete discussion of well options is in Chap. 7 of thisbook and Chap. 7 (pp. 74–86) of M-13.

Problems — Chapter 1

1. Model selection. The reservoir shown below is sealed bypinchouts. It is undersaturated (above bubble point pressure)

and you would expect a depletion drive mechanism from thedata available.

It has been proposed to convert one of the four present pro-

ducers to an injection well to maintain pressure above the bubble




point. You have been told to evaluate the reservoir engineeringaspects of the proposal.

What tools would you use (type of model, simple calcula-tions, etc.) and what would you expect to learn from each?

Which tool would you use first? Which last? How would you

grid the field areally for a study?

2. Length calculations. In an areal model (using aerial welllocations and grids), two cells will be one mile apart (5,280 ft).

The reservoir dips continuously at an angle of 6◦. What is the

x

r

θ

cosθ = x/r

y



Introduction 45

true length between the two cells and how much error have weintroduced by using the areal map?

3. Porosity. A reservoir is discovered at 5,000 psig having alog-measured porosity of 20%. If the abandonment pressure is

1,000 psig, what value of porosity will exist at abandonment?The formation compressibility is 3.6 microsips (3.6× 10−6/psi).

If the porosity of 20% had been measured in a core analysis

laboratory at a pressure of 100 psig, what would the value of theoriginal reservoir porosity be?

4. Permeability averaging. Calculate the horizontal and ver-

tical permeabilities for the reservoir shown; also, calculate the

geometric-average permeability.

5. Dip angle. Calculate the change in elevation over 3 miles fora reservoir having an 8◦ dip angle. Also determine the resulting

additional pressure that would exist at this depth (use a brine

hydraulic gradient of 0.466 psi/ft).




6. Pressure gradients. The pressure at the gas–oil contact is2,200 psia in a reservoir; the contact is very sharp and no transi-

tion zone exists. What pressure would occur 40′ below the GOC?The oil has a reservoir density of 52.1 lb/ft3.

7. Laboratory-determined capillary pressure. Calculate the

height of a water–oil transition zone in a reservoir having a criti-cal water saturation of 35%; the laboratory air–water capillarity

at the critical water saturation is 18 psi and the air–water inter-facial tension is 72 dynes/cm. The stock tank density of the

crude is 35◦ API and it has a water–oil interfacial tension of 24

dynes/cm at reservoir conditions. The water specific gravity is1.09 and the formation volume factors for water and oil are 1.02

and 1.24 RVB/STB, respectively. The solution gas is 540 andthe dissolved gas gravity is 0.7.

8. Estimation of capillary pressures using the J-function.

Estimate, using the J-function, the capillary pressures for a

00

0.5

1.0

J(Sw)

1.5

20 40Sw (%)

60 80 100



Introduction 47

reservoir having a critical water saturation of 22% and an oil–

water interfacial tension of 27 dynes/cm. The oil has a gravity of28◦ API and the water has a specific gravity of 1.1. The reservoir

averages 15% porosity and has a 130 md permeability. Calculate

the capillary pressures at saturations of 22, 25, 30, 50, 80, 100%.

9. Gas–oil capillary pressure. Calculate the capillary pressureat the top of a GOC consisting of a 5′ transition zone. Oil den-

sity is 52 lb/ft3 at stock tank conditions (38◦ API) and the gasgravity is 0.75. Formation volume factors for oil and gas are 1.20

RVB/STB and 0.80 RVB/MCF. Solution gas is 0.7.

reservoir simulation, 5775_chap01

Documents